目的 针对变分方法在去除图像乘性噪声时易产生“阶梯效应”的问题,分析研究了几种经典乘性去噪变分模型的特性和相关性,在此基础上考虑到分数阶微分的频率特性,提出一种用于去除乘性Gamma噪声的分数阶凸变分模型。方法 提出的分数阶凸变分模型是经典I-divergence变分模型的分数阶扩展。基于对偶理论,提出一种用于求解该模型的分数阶原始对偶算法。并且基于鞍点理论,分析了确保算法收敛的参数取值范围。结果 实验中从频域角度分析并验证了提出的分数阶变分模型较经典的一阶变分模型能够有效缓解“阶梯效应”现象,更好地保持图像的中频纹理和高频边缘信息。同时提出的分数阶原始对偶数值算法能有效收敛,且收敛速度较快。结论 本文提出了一种去除图像乘性噪声的分数阶变分模型,该模型可采用一种基于预解式的原始对偶算法求解。实验结果表明,提出的模型能有效改善图像的视觉效果,采用的数值算法能有效快速收敛。

Objective Considering that the variation methods for image multiplicative noise removal exhibit the staircase effect problem, we analyze the characteristic and correlation of several classical multiplicative denoising variation models. With the frequency characteristic of the fractional differential considered, a fractional-order convex variation model for multiplicative Gamma noise removal is proposed.Method Our fractional-order convex variation model is the fractional-order generalization of the classical I-divergence variation model. Based on duality theory, a fractional-order primal-dual algorithm to solve the model is proposed. The range of the parameter is given according to saddle-point theory to guarantee algorithm convergence. Result In terms of frequency domain aspects, the experiments verify that the proposed fractional-order variation model is effective in relaxing the staircase effect and preserving medium-frequency texture information in a cardiac ultrasound image, as well as high-frequency building edges information in the “Cameraman” image compared with the classical first-order variation model. The proposed fractional-order primal-dual algorithm can also effectively converge and exhibits a fast convergence speed.Conclusion To produce denoised images with minimal loss of image details, this study proposes a fractional-order variation model for image multiplicative noise removal. Classical variation numerical algorithms need to compute the derivative of a non-differentiable function (i.e., the fractional-order regulation term), so we refer to a primal-dual algorithm based on a resolvent for an alternative solution. Experiment results indicate that the proposed model can effectively improve the image visual effect, and the adoptive numerical algorithm demonstrates a fast convergence speed.